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Inertia theorems for matrices, controllability, and linear vibrations
LINEAR
ALGEBRA
AND
ITS
APPLICATIONS
Inertia Theorems for Matrices, and Linear Vibrations HARALD
K. WIMMER
Technische
Hochschule
8, 337-343
337
(1974)
Controllability,
Graz
Graz, Austria
Recommended
by Hans
Schneider
ABSTRACT If H is a Hermitian has as many
eigenvalues
eigenvalues
[5].
the rank
and
W = AH
+ HA * is positive
with positive
(negative)
real part
matrix
Theorems
of this type
of the controllability
theorem.
As an application,
equation
iV% + (D + G)k
matrix a result + Kx
=
are known of A and in [8] and
definite,
as H has positive
as inertia
theorems.
W is used
to derive
[4] on a damping
then
A
(negative) In this note
a new inertia problem
of the
0 is extended.
1. INTRODUCTION
The inertia of an n x n matrix A with complex elements is defined as the integer triple In A = (n(A), Y(A), 6(A)), where n(A), V(A), and 6(A) are, respectively, the number of eigenvalues of A with positive, negative and zero real part. Generalizing a theorem of Lyapunov, Ostrowski, and Schneider [5] and Taussky [7] relate the inertia of a matrix to the matrix inequality AH + HA* > 0 (positive definite). H shall always denote a Hermitian matrix. INERTIA
THEOREM
[5].
If AH + HA* > 0, lhen In A = In H ad
6(A) = d(H) = 0. The case AH + HA* > 0 (positive semidefinite) is discussed Carlson and Schneider [I]. We note the following result. 0 American
Elsevier
Publishing
Company,
Inc.,
by
1974
338
HARALDK.WIMMER 1 [l]. 1f H is nonsingular
THEOREM
and A has no eigenvalues
on
axis (i.e., 6(A) = 6(H) = 0), then AH + HA* > 0 implies
the imaginary
In A = In H. In this note we use the concept of a controllable pair of matrices to investigate the inertia of A. Our results will be applied to the equation MZ + (D + G)3i + Kx = 0 and will provide an extension of a theorem of Zajac [8] and Mtiller [4]on the pervasive damping of linear mechanical systems. 2,SOMELEMMAS
OFCONTROLTHEORY
For the basic definitions and lemmas of linear control theory in this section we refer to [2] and [3]. Let B be a complex n x Y matrix. The controllability matrix C(A IB) of A and B is defined as the n x nr matrix C(AjB)
= (B, AB, A2B,. . ., An-lB).
The pair (A, B) is called controllable, 1 [2].
LEMMA
if rank C(A(B)
= n.
(A, B) is controllable, if and only if
rank(A
-
ill, B) = n
for each eigenvalue A of A. 2 [3,p. 991.
LEMMA
Given the pair (A, B) there exists a nonsingular
S such that S-lB=
and (A,,, LEMMA pair
B,) is controllable;
B1 (10'
rank C(A JB) = rank C(A1ljB1).
3 [2].If (A, B) is controllable, then for every
Y
x n matrix C the
(A + BC, B) is controllable.
3.INERTIATHEOREMS THEOREM
2. If AH + HA*
= W, W >, 0 and (A, W) is controllable,
then 6(A) = d(H) = 0 and In A =
In H.
INERTIA
AND
Proof.
339
CONTROLLABILITY
We show by contradiction
that our assumptions
imply- d(A) = 0
and \Hl # 0. In A = In H follows from Theorem 1. Let A have an eigenvalue ice with zero real part and let 2t* be a corresponding left eigenvector. Then u*A = ixu* and u*(ilH + HA*)u = u*Wu = 0. TV > 0 implies u*W = 0. Thus there exists a zt # 0 such By Lemma 1 (A, W) is not that u*(A - ial) = 0 and zt*W = 0. controllable. We assume now that H is singular, so there is a “J # 0 with Then v*AkWA*% = v*H = 0. Suppose v*A”H = 0 for some k. v*A”(AH + HA*)A% = 0 and again TV 3 0 implies v*A”W = 0. We prove by induction v*A”H
= 0,
k = 0, 1, 2,. . . .
(1)
By the assumption on II (1) is true for k = 0. If (1) holds for k = Y, then From (1) we deduce 0 = v*A’W = ti*A’+lH + v*A’HA* = v*Ar+lH. by the argument above v*A”W = 0 for k 3 0 or w*(W, AW,.
. ., An-lW)
which means (A, W) is not controllable. The following theorem is due to Snyders In this paper a different proof is given. THEOREM 3.
= 0
and Zakai [6, Corollary
4.11.
If
AH+HA*= thelz v(A) = 0, 6(A) = p(A)
W,
W 3 0
and
= n -
rank C(AjW),
H > 0, n(A)
where p(A) is the number of elementary divisors of purely
(2)
= rank C(AIW) imagimary eigen-
values of A. Proof. From [l] we know that v(A) = 0 and 6(A) = p(A). Without loss of generality we can assume A and Win (2) to be partitioned according to Lemma 2: A=(A;l
;:I),
where (A,,, IV,,) is controllable. rank C(A,,IW,,) = rank C(A/W).
-=(;I
:),
W,,>O,
If Al1 is an s x s matrix, then s = Let H > 0 be partitioned conformably
HARALD
340
K. WIMMER
Then
H into block-diagonal
transforms
form
QHQ* = where HI1 = H,, -
H,,H,‘H;
= r;’
QAQ-’
r;’ i2,)>
> 0.
k”),
QwQ*
=(;;;
72).
22
From Q(AH + HA*)Q* we get the equations
= (QAQ-l)(QHQ*)
+ (QHQ*)(QAQ-l)*
Ai&,
+ &A:
= W,,,
&,
> 0,
Aa&22
+ HzzA:
= 0,
H,2 > 0.
Wir 3 0,
= QWQ*
(3) (4)
In (3) the conditions of Theorem 2 are satisfied, therefore d(A,,) = 0. Because of Hll > 0 all eigenvalues of Al1 are in the right half plane: v(All) = 0, n(A,,) = rank C(A1W). In (4) H2a > 0 implies A22 is similar to a skew-Hermitian matrix, hence all eigenvalues of A,, are purely imaginary: v(Aa2) = n(A,,) = 0, (S(A,,) = $(A,,) = n - rank C(AjW). COROLLARY
1.
If
AH + HA*
= W,
W 3 0 and
H > 0, then all
eigenvalues of A have positive real part, if and only if the pair (A, W) is controllable. 4. LINEAR
VIBRATIONS
Consider the equation Mli
+ (D +
G)a+ Kx =
0,
(5)
where all matrices are n x n and real and where M, D, K are symmetric and G is skew-symmetric. Let M be nonsingular and D 3 0. We associate with (5) the matrix F(A) = A2M + i(D + G) + K.
(6)
INERTIA
AND
The latent Nontrivial
341
CONTROLLABILITY
roots of (6) are defined as the values of 1 for which /F(A)1 = 0. solutions of F(il)q = 0 or yTF(jl) = 0 are known as latent
vectors. Equation
(5) is equivalent
to the first order equation
j, = Ay with
I -
The eigenvalues
M-lK
-
of A are the latent
Mwl(D
+ G)
roots of F(I).
We put
then f@ < 0.
A”TV + Vk = IV,
(7)
Applying Theorem 2 to (7) (with trivial change of sign) we generalize of Zajac [8] and Mtiller [4]. THEOREM
imaginary
4.
If the pair
(XT, I&‘) is controllable,
latent roots, the number
real part is equal to n(M)
oflated
+ n(K),
a result
then F(1) has no Purely
roots with negative,
resp. positive,
Yes@. Y(M) + y(K).
Because of the given block structure of k and #’ criteria for controllability of (AT, m) can be simplified. We also mention that (AT, m) to be controllable means (A, @) to be observable [3]. LEMMA
4.
The follozeling statements
(AmT, %‘) is controllable.
(a) (b)
(‘4, D) is controllable
(c)
For each latent root of Z(a)
where
rank(Z(A), Proof.
aye equivalent:
From Lemma
= A2M -
;1G + K
ID) = n.
3 we infer (a) o (b), since
342
HARALD
K. WIMMER
An easy calculation shows that the eigenvalues of A are the latent roots of Z(1) and that rank(ri^ - II, b) = n + rank(Z(A), AD). Thus (b) o (c) by Lemma 2. We consider (5) now under the additional assumptions M > 0 and K > 0. If D > 0, then all roots of F(A) necessarily lie in the left half plane and the equilibrium x(t) E 0 of (5) is asymptotically stable. If D 3 0 is singular, then (5) may have periodic solutions (which means F(1,) has purely imaginary roots). The damping - L)i is called pervasive, if it acts on all components of X, so that all solutions of (5) tend to zero as t ---, co. THEOREM
-
5
[4].
If
Di in (5) is pervasive,
M > 0, K > 0 and D > 0, then the damping if and only if the pair (A^,h) is controllable.
Proof. Because of M > 0 and K > 0 the matrix V in (7) is positive definite. The theorem is an immediate consequence of Corollary 1 and Lemma 4. We observe that for pervasive damping we need not require M to be nonsingular. THEOREM 6.
roots of F(I)
If M > 0, D 3 0, K > 0 and GT = -
G, then all latent
= A2M + A(D + G) + K have negative real part, if and only
if there is no latent vector q of A2M + ;iG + K with Dq = 0. Proof. Let Jo be a latent value of F(i), then the assumptions imply il,, # 0 and Re lo < 0. Suppose F(A) has an imaginary root icr, c( # 0 and F(ia)q = 0, q # 0. Then 0 = qTF(ia)q = - Cr2qTMq+ qTKq + iccqTDq. Separating real and imaginary parts we obtain Dq = 0 and (- x2M + z&G + K)q = 0. The converse follows from the fact that all latent roots of 12M + 1G + K lie on the im,aginary axis. The author is grateful to Dr. P. Ch. Miiller for helpful comments. REFERENCES 1 D. Carlson and H. Schneider,
Inertia theorems:
the semidefinite
case, J. Math.
Anal. Appl. 6(1963), 430-446. 2 M. L. J. Hautus, Controllability and observability conditions for linear autonomous systems, Nederl. Akad. Wet. Proc. A72(1969), 443-448. 3 E. B. Lee and L. Markus, Foundations (1967).
of Optimal
Co&o1
Theory,
Wiley, New York
INERTIA
AND CONTROLLABILITY
4 P. C. Miiller, Asymptotische positiv
semidefiniter
343
Stabilitlt
Dampfungsmatrix,
van linearen mechanischen Z. Angew.
Math.
Mech.
Systemen 61(1971),
mit
T197-
T198. 5 A. Ostrowski and H. Schneider, Some theorems on the inertia of general matrices, J. Math.
Anal.
Appl.
4(1962),
72-84.
6 J. Snyders and M. Zakai, On nonnegative -
C, SIAM
J. A&%.
Math.
7 0. Taussky, A generalization
solutions of the equation
AD + DA’
=
18(1970), 704-714. of a theorem by Lyapunov,
J. Sot. Ind. Appl.
Math.
9(1961), 640-643. 8 E. E. Zajac, Comments on “Stability extension, J. AlAA Received
November
3(1965), 7, 1972;
of damped mechanical systems”
1749-1750. revised
February,
1973
and a further
ALGEBRA
AND
ITS
APPLICATIONS
Inertia Theorems for Matrices, and Linear Vibrations HARALD
K. WIMMER
Technische
Hochschule
8, 337-343
337
(1974)
Controllability,
Graz
Graz, Austria
Recommended
by Hans
Schneider
ABSTRACT If H is a Hermitian has as many
eigenvalues
eigenvalues
[5].
the rank
and
W = AH
+ HA * is positive
with positive
(negative)
real part
matrix
Theorems
of this type
of the controllability
theorem.
As an application,
equation
iV% + (D + G)k
matrix a result + Kx
=
are known of A and in [8] and
definite,
as H has positive
as inertia
theorems.
W is used
to derive
[4] on a damping
then
A
(negative) In this note
a new inertia problem
of the
0 is extended.
1. INTRODUCTION
The inertia of an n x n matrix A with complex elements is defined as the integer triple In A = (n(A), Y(A), 6(A)), where n(A), V(A), and 6(A) are, respectively, the number of eigenvalues of A with positive, negative and zero real part. Generalizing a theorem of Lyapunov, Ostrowski, and Schneider [5] and Taussky [7] relate the inertia of a matrix to the matrix inequality AH + HA* > 0 (positive definite). H shall always denote a Hermitian matrix. INERTIA
THEOREM
[5].
If AH + HA* > 0, lhen In A = In H ad
6(A) = d(H) = 0. The case AH + HA* > 0 (positive semidefinite) is discussed Carlson and Schneider [I]. We note the following result. 0 American
Elsevier
Publishing
Company,
Inc.,
by
1974
338
HARALDK.WIMMER 1 [l]. 1f H is nonsingular
THEOREM
and A has no eigenvalues
on
axis (i.e., 6(A) = 6(H) = 0), then AH + HA* > 0 implies
the imaginary
In A = In H. In this note we use the concept of a controllable pair of matrices to investigate the inertia of A. Our results will be applied to the equation MZ + (D + G)3i + Kx = 0 and will provide an extension of a theorem of Zajac [8] and Mtiller [4]on the pervasive damping of linear mechanical systems. 2,SOMELEMMAS
OFCONTROLTHEORY
For the basic definitions and lemmas of linear control theory in this section we refer to [2] and [3]. Let B be a complex n x Y matrix. The controllability matrix C(A IB) of A and B is defined as the n x nr matrix C(AjB)
= (B, AB, A2B,. . ., An-lB).
The pair (A, B) is called controllable, 1 [2].
LEMMA
if rank C(A(B)
= n.
(A, B) is controllable, if and only if
rank(A
-
ill, B) = n
for each eigenvalue A of A. 2 [3,p. 991.
LEMMA
Given the pair (A, B) there exists a nonsingular
S such that S-lB=
and (A,,, LEMMA pair
B,) is controllable;
B1 (10'
rank C(A JB) = rank C(A1ljB1).
3 [2].If (A, B) is controllable, then for every
Y
x n matrix C the
(A + BC, B) is controllable.
3.INERTIATHEOREMS THEOREM
2. If AH + HA*
= W, W >, 0 and (A, W) is controllable,
then 6(A) = d(H) = 0 and In A =
In H.
INERTIA
AND
Proof.
339
CONTROLLABILITY
We show by contradiction
that our assumptions
imply- d(A) = 0
and \Hl # 0. In A = In H follows from Theorem 1. Let A have an eigenvalue ice with zero real part and let 2t* be a corresponding left eigenvector. Then u*A = ixu* and u*(ilH + HA*)u = u*Wu = 0. TV > 0 implies u*W = 0. Thus there exists a zt # 0 such By Lemma 1 (A, W) is not that u*(A - ial) = 0 and zt*W = 0. controllable. We assume now that H is singular, so there is a “J # 0 with Then v*AkWA*% = v*H = 0. Suppose v*A”H = 0 for some k. v*A”(AH + HA*)A% = 0 and again TV 3 0 implies v*A”W = 0. We prove by induction v*A”H
= 0,
k = 0, 1, 2,. . . .
(1)
By the assumption on II (1) is true for k = 0. If (1) holds for k = Y, then From (1) we deduce 0 = v*A’W = ti*A’+lH + v*A’HA* = v*Ar+lH. by the argument above v*A”W = 0 for k 3 0 or w*(W, AW,.
. ., An-lW)
which means (A, W) is not controllable. The following theorem is due to Snyders In this paper a different proof is given. THEOREM 3.
= 0
and Zakai [6, Corollary
4.11.
If
AH+HA*= thelz v(A) = 0, 6(A) = p(A)
W,
W 3 0
and
= n -
rank C(AjW),
H > 0, n(A)
where p(A) is the number of elementary divisors of purely
(2)
= rank C(AIW) imagimary eigen-
values of A. Proof. From [l] we know that v(A) = 0 and 6(A) = p(A). Without loss of generality we can assume A and Win (2) to be partitioned according to Lemma 2: A=(A;l
;:I),
where (A,,, IV,,) is controllable. rank C(A,,IW,,) = rank C(A/W).
-=(;I
:),
W,,>O,
If Al1 is an s x s matrix, then s = Let H > 0 be partitioned conformably
HARALD
340
K. WIMMER
Then
H into block-diagonal
transforms
form
QHQ* = where HI1 = H,, -
H,,H,‘H;
= r;’
QAQ-’
r;’ i2,)>
> 0.
k”),
QwQ*
=(;;;
72).
22
From Q(AH + HA*)Q* we get the equations
= (QAQ-l)(QHQ*)
+ (QHQ*)(QAQ-l)*
Ai&,
+ &A:
= W,,,
&,
> 0,
Aa&22
+ HzzA:
= 0,
H,2 > 0.
Wir 3 0,
= QWQ*
(3) (4)
In (3) the conditions of Theorem 2 are satisfied, therefore d(A,,) = 0. Because of Hll > 0 all eigenvalues of Al1 are in the right half plane: v(All) = 0, n(A,,) = rank C(A1W). In (4) H2a > 0 implies A22 is similar to a skew-Hermitian matrix, hence all eigenvalues of A,, are purely imaginary: v(Aa2) = n(A,,) = 0, (S(A,,) = $(A,,) = n - rank C(AjW). COROLLARY
1.
If
AH + HA*
= W,
W 3 0 and
H > 0, then all
eigenvalues of A have positive real part, if and only if the pair (A, W) is controllable. 4. LINEAR
VIBRATIONS
Consider the equation Mli
+ (D +
G)a+ Kx =
0,
(5)
where all matrices are n x n and real and where M, D, K are symmetric and G is skew-symmetric. Let M be nonsingular and D 3 0. We associate with (5) the matrix F(A) = A2M + i(D + G) + K.
(6)
INERTIA
AND
The latent Nontrivial
341
CONTROLLABILITY
roots of (6) are defined as the values of 1 for which /F(A)1 = 0. solutions of F(il)q = 0 or yTF(jl) = 0 are known as latent
vectors. Equation
(5) is equivalent
to the first order equation
j, = Ay with
I -
The eigenvalues
M-lK
-
of A are the latent
Mwl(D
+ G)
roots of F(I).
We put
then f@ < 0.
A”TV + Vk = IV,
(7)
Applying Theorem 2 to (7) (with trivial change of sign) we generalize of Zajac [8] and Mtiller [4]. THEOREM
imaginary
4.
If the pair
(XT, I&‘) is controllable,
latent roots, the number
real part is equal to n(M)
oflated
+ n(K),
a result
then F(1) has no Purely
roots with negative,
resp. positive,
Yes@. Y(M) + y(K).
Because of the given block structure of k and #’ criteria for controllability of (AT, m) can be simplified. We also mention that (AT, m) to be controllable means (A, @) to be observable [3]. LEMMA
4.
The follozeling statements
(AmT, %‘) is controllable.
(a) (b)
(‘4, D) is controllable
(c)
For each latent root of Z(a)
where
rank(Z(A), Proof.
aye equivalent:
From Lemma
= A2M -
;1G + K
ID) = n.
3 we infer (a) o (b), since
342
HARALD
K. WIMMER
An easy calculation shows that the eigenvalues of A are the latent roots of Z(1) and that rank(ri^ - II, b) = n + rank(Z(A), AD). Thus (b) o (c) by Lemma 2. We consider (5) now under the additional assumptions M > 0 and K > 0. If D > 0, then all roots of F(A) necessarily lie in the left half plane and the equilibrium x(t) E 0 of (5) is asymptotically stable. If D 3 0 is singular, then (5) may have periodic solutions (which means F(1,) has purely imaginary roots). The damping - L)i is called pervasive, if it acts on all components of X, so that all solutions of (5) tend to zero as t ---, co. THEOREM
-
5
[4].
If
Di in (5) is pervasive,
M > 0, K > 0 and D > 0, then the damping if and only if the pair (A^,h) is controllable.
Proof. Because of M > 0 and K > 0 the matrix V in (7) is positive definite. The theorem is an immediate consequence of Corollary 1 and Lemma 4. We observe that for pervasive damping we need not require M to be nonsingular. THEOREM 6.
roots of F(I)
If M > 0, D 3 0, K > 0 and GT = -
G, then all latent
= A2M + A(D + G) + K have negative real part, if and only
if there is no latent vector q of A2M + ;iG + K with Dq = 0. Proof. Let Jo be a latent value of F(i), then the assumptions imply il,, # 0 and Re lo < 0. Suppose F(A) has an imaginary root icr, c( # 0 and F(ia)q = 0, q # 0. Then 0 = qTF(ia)q = - Cr2qTMq+ qTKq + iccqTDq. Separating real and imaginary parts we obtain Dq = 0 and (- x2M + z&G + K)q = 0. The converse follows from the fact that all latent roots of 12M + 1G + K lie on the im,aginary axis. The author is grateful to Dr. P. Ch. Miiller for helpful comments. REFERENCES 1 D. Carlson and H. Schneider,
Inertia theorems:
the semidefinite
case, J. Math.
Anal. Appl. 6(1963), 430-446. 2 M. L. J. Hautus, Controllability and observability conditions for linear autonomous systems, Nederl. Akad. Wet. Proc. A72(1969), 443-448. 3 E. B. Lee and L. Markus, Foundations (1967).
of Optimal
Co&o1
Theory,
Wiley, New York
INERTIA
AND CONTROLLABILITY
4 P. C. Miiller, Asymptotische positiv
semidefiniter
343
Stabilitlt
Dampfungsmatrix,
van linearen mechanischen Z. Angew.
Math.
Mech.
Systemen 61(1971),
mit
T197-
T198. 5 A. Ostrowski and H. Schneider, Some theorems on the inertia of general matrices, J. Math.
Anal.
Appl.
4(1962),
72-84.
6 J. Snyders and M. Zakai, On nonnegative -
C, SIAM
J. A&%.
Math.
7 0. Taussky, A generalization
solutions of the equation
AD + DA’
=
18(1970), 704-714. of a theorem by Lyapunov,
J. Sot. Ind. Appl.
Math.
9(1961), 640-643. 8 E. E. Zajac, Comments on “Stability extension, J. AlAA Received
November
3(1965), 7, 1972;
of damped mechanical systems”
1749-1750. revised
February,
1973
and a further